User:Boris Tsirelson/sandbox1

$\newcommand{\Om}{\Omega} \newcommand{\F}{\mathcal F} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ A probability space is called standard if it is a standard Borel space endowed with a probability measure, completed with null sets, and possibly augmented with another null set. (See Definition 1 below.) Every standard probability space is isomorphic (mod 0) to an interval with Lebesgue measure, a finite or countable set of atoms, or a combination of both. (See Theorem ? below.)

Example. The set of all continuous functions $[0,\infty)\to\R$ with the Wiener measure is a standard probability space.

Non-example. The set $[0,1]^\R$ of all functions $\R\to[0,1]$ with the product of Lebesgue measures is a nonstandard probability space.

Definition 1. A probability space $(\Om,\F,P)$ is standard if it is complete (that is, $\F$ contains all null sets; these are sets $A\subset\Om$ such that $ \exists B\in\F \big( A\subset B \land P(B)=0 \big) $) and there exist a subset $ \Om_1\subset\Om$ and a σ-field (in other words, σ-algebra) $\B$ on $\Om_1$ such that $(\Om_1,\B)$ is a standard Borel space and $\forall A\in\F \;\; \exists B\in\B \; \big( B \subset A \land P(B)=P(A) \big)$.

(See [S, p. 170], [D, Sect. 11.5].)

Definition 3. A measurable space is called universally measurable if it is isomorphic to some universally measurable metric space (as defined above) with the Borel σ-algebra. (See [S, p. 171].)

Thus, the phrase "universally measurable space" is ambiguous; it can be interpreted as "universally measurable metric space" or "universally measurable measurable space"! The latter can be replaced with "universally measurable Borel space", but the ambiguity persists. Fortunately, the ambiguity is rather harmless by the following result.

On terminology

In [M, Sect. 6] universally measurable spaces are called metrically standard Borel spaces.

In [K, Sect. 21.D] universally measurable subsets of a standard (rather than arbitrary) measurable space are defined.

In [N, Sect. 1.1] an absolute measurable space is defined as a separable metrizable topological space such that every its homeomorphic image in every such space (with the Borel σ-algebra) is a universally measurable subset. The corresponding measurable space (with the Borel σ-algebra) is also called an absolute measurable space in [N, Sect. B.2].

References

[I]	Kiyosi Itô, "Introduction to probability theory", Cambridge (1984).   MR0777504   Zbl 0545.60001
[C]	Donald L. Cohn, "Measure theory", Birkhäuser (1993).   MR1454121   Zbl 0860.28001
[D]	Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989).   MR0982264   Zbl 0686.60001
[M]	George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[K]	Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[N]	Togo Nishiura, "Absolute measurable spaces", Cambridge (2008).   MR2426721   Zbl 1151.54001